Note: Last night’s homework solutions appear in the In The Classroom file.
Warmup #1 requires students to reason abstractly and quantitatively (MP2) in finding the constant k that “connects” the two pieces of the piecewise function. Students likely will approach this problem algebraically, but there is value in students viewing and solving this problem graphically. As I circulate around the room, I will ask students to think of a non-algebraic way to check the reasonableness and correctness of their algebraic solution.
Warm-up #3 motivates the use of the Mean Value Theorem, which will be reviewed and applied as part of today’s lesson. I used a cubic function in factored form to provide students an opportunity to consider ways to make their algebraic work easier, prior to differentiating the function. If my students mindlessly set out to multiply the first two binomial factors (x – 6) and (x – 1) from left to right, their next step of multiplying the third binomial factor will involve a trinomial multiplied by a binomial, making the algebra unnecessarily long and tedious. But if students analyze the function first, employ the commutative property of multiplication, and recognize that (x – 1) and (x + 1) multiplies into a difference of squares (x2 – 1) with only two terms, then subsequently multiplying by the (x – 6) term becomes simpler. It is difficult to break students’ habit of rushing to perform the first mathematical procedure that comes to mind. With frequent opportunities like this one to analyze the function, consider its structure (MP6), and then begin algebraic work, students will eventually learn to ask whether there are more efficient methods before beginning to solve the problem.
On last night’s HW set, students often have questions about the conical tank problem’s setup and solution. The difficult part is how to setup and use similar triangles to relate the radius and height variables, and then using that relationship to substitute out one of the variables in the Volume function prior to differentiating. It often helps my students to think of the similar triangles ratio between the radius and height as a “constraint”, akin to optimization problems where the constraint is also used to substitute out one of the variables. The In The Classroom file has an additional example if students need more practice after your class discussion about the homework problem.
The body of today’s lesson may be split into two parts:
To begin our review of integration, I project for students the activity sheet from the 1st week of school where a student goes to grandma’s house for lunch. On the back side students discovered that the area under the velocity graph ?coincidentally? gives the distance that the student traveled – technically displacement as students now know the difference between them. Using dimensional analysis (see Lunch at Grandma's Graph), we can justify why the rectangular areas under the velocity curve yield a measure of position. In the process, students attend to precision (MP 6) by carrying units through their calculations to interpret the units of the area output.
Once students are convinced of the purpose of fitting rectangles under functions, it follows naturally to try to make the rectangles fit better and better under curvy functions, not the simple line segments from the Introduction to Calculus sheet. Rather than me continuing the lecture at this point, I ask students to do a quick-write to recall the specific details for how we improved our area approximations under curvy functions. If students struggle to recall this work from earlier in the semester then direct them to collaborate with their neighbors and even look up Riemann Sums in their notes before completing their quick-writes and sharing with the whole class.
I hope that my students will recall two equivalent approaches: 1) increasing the number of rectangles to infinity that are fit into a finite interval, and 2) reducing the width of each of those rectangles, dx, to zero (dx à 0). This second approach motivates the area formula for one rectangle , and summing all rectangles (we now call this integrating) gives the familiar. After this discussion I play (or replay if used earlier in the semester) the Matheatre song Without Riemann because the lyrics emphasize the concept of shrinking the widths of each rectangle to zero as a way to improve the accuracy of the Riemann Sums. Plus, students get a kick out of corny math songs!
I expect that my students will be able to apply their understanding of integration to functions defined algebraically (very familiar to students), graphically (less familiar), and tabularly (very unfamiliar).
The In The Classroom file includes task where a function is provided by giving several points in a table. The task asks students to:
I plan to give my students plenty of time to figure out how to answer (the second question in particular). I want my students to understand that they can still make estimates about derivatives and integrals given individual points only and not the entire equation of a function.
Although we do not know how to “connect the dots” based on the points given in the table, we can still compute LRAM or RRAM as estimates of the definite integral. (In fact, this approach can be very helpful for students’ Car Dashboard Projects!) Using the slope between two nearby points is a viable strategy for approximating the slope of the tangent line at a point somewhere between the two endpoints. Revisiting the image from Problem 3 on today's Warmup is a nice way to wrap up this portion of the lesson and help students understand the Mean Value Theorem by connecting the tabular and graphical representations.
Today's Closure Activity: Write a note to an absent student explaining when RRAM and LRAM overestimate or underestimate the true area under a function. Include a picture with your written explanation.
F - read TEXTBOOK examples, assign problems about MVT and basics of multiple derivatives (inc/dec, concave up/down, inflection points)
I - 1993 #18
V - read TEXTBOOK examples, assign problems about Riemann sums algebraically, graphically, and tabularly
E - Set 5: 1969 #9, 1973 #26, 1985 #31 – submit online